Use of Elements of Large Size
In all the foregoing analyses, it is assumed that the finite element is small enough for the resulting stress – strain curve to be stable. If the element is too large, then the resulting stress-strain curve has a snapback, as shown in Fig. 8.3.6d. If this occurs, the finite element analysis will be very difficult to stabilize and will dissipate more energy than it actually should. It may be argued that the problem should be solved by using smaller elements, but this may be computationally too expensive and it may be worth using larger elements if the accuracy is not greatly sacrificed.
To simplify the problem, let us consider the simple linear softening depicted in Fig. 8.6.5a for the actual crack band thickness hc. If the element size is h > hc, the stress-strain curve for the element is as shown in Fig. 8.6.5b. The softening branch becomes vertical when Єhc/h — f’t/E (see Fig. 8.6.5c), i. e., for h = hcecE//(‘. Since in this linear case Gf = hcf’tei/2, eliminating hc leads to the simple condition
h = Uch (8.6.15)
Thus, for h > 2(ch, a snapback occurs as shown in Fig. 8.6.5d. Because in the finite element computation the nodal displacements are controlled, the stress will drop to zero as soon as the peak is reached and the dissipated energy will appear to be the area OPB instead of the area OP A which is the correct value. This means that using larger elements will make the material appear tougher than it actually is.
A solution to this problem (Bazant 1985b, c) is to replace the actual stress-strain curve with snapback by a stress-strain diagram of the same area having a vertical stress drop (Fig. 8.6.5e), To keep the same area, one must reduce the tensile strength from // to feq so that
Thus, the strength must be reduced in inverse proportion to the square root of the element size. For the case of vertical stress drop, the fracture process zone has the smallest length permitted by the finite element subdivision. Therefore, this represents the closest possible approximation to LEFM. Since the element size is normally taken proportional to the structure size, this means that the crack band model with a vertical drop yields an approximate equivalent LFFM behavior for structures of large sizes. However,
this is not the only way to handle the problem of brittle behavior with large elements. In the following, an energy-based analysis is presented as a possible alternative.